Provably Accurate ODE Forecasting Through Explicit Trajectory Optimization

Helmuth Naumer, Farzad Kamalabadi

Primary Area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

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Keywords: Forecasting, Dynamical Systems, Parameter Estimation, Bayesian Inference

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Abstract:

This work introduces a method to enable accurate forecasting of time series governed by ordinary differential equations (ODE) through the usage of cost functions explicitly dependent on the future trajectory rather than the past measurement times. We prove that the space of solutions of an $N$-dimensional, smooth, Lipschitz ODE on any given finite time horizon is an $N$-dimensional Riemannian manifold embedded in the space of square integrable continuous functions. This finite dimensional manifold structure enables the application of common statistical objectives such as maximum likelihood (ML), maximum a posteriori (MAP), and minimum mean squared error (MMSE) estimation directly in the space of feasible ODE solutions. The restriction to feasible trajectories of the system limits known issues such as oversmoothing seen in unconstrained MMSE forecasting. We demonstrate that direct optimization of trajectories reduces error in forecasting when compared to estimating initial conditions or minimizing empirical error. Beyond theoretical justifications, we provide Monte Carlo simulations evaluating the performance of the optimal solutions of six different objective functions: ML, MAP state estimation, MMSE state estimation, MAP trajectory estimation, MMSE trajectory estimation over all square integrable functions, and MMSE trajectory estimation over solutions of the differential equation.

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Submission Number: 5850